Hilbert Space Research Articles

Mini-Workshop: Standard Subspaces in Quantum Field Theory and Representation Theory

Real standard subspaces of complex Hilbert spaces are long known to provide the right language for Tomita–Takesaki modular theory of von Neumann algebras. In recent years they have also become an object of prominent interest in mathematical quantum field theory (QFT) and unitary representation theory of Lie groups. This workshop brought together mathematicians and physicists working with standard subspaces, particularly in QFT (construction of QFT models, characterization of entropy, information-theoretic aspects), nets of standard subspaces on causal homogeneous spaces and aspects of reflection positivity and euclidean models related to standard subspaces and modular theory.

On the relations between Auerbach or almost Auerbach Markushevich systems and Schauder bases

We establish that the summability of the series ∑εn is the necessary and sufficient criterion ensuring that every (1+εn)-bounded Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if nεn→∞, then in ℓ2 there exists a (1+εn)-bounded Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis.

Consistent Sampling Approximations in Abstract Hilbert Spaces

This paper considers generalized consistent sampling and reconstruction processes in an abstract separable Hilbert space. Using an operator-theoretical approach, quasi-consistent and consistent approximations with optimal properties, such as possessing the minimum norm or being closest to the original vector, are derived. The results are illustrated with several examples.

How to Partition a Quantum Observable.

We present a partition of quantum observables in an open quantum system that is inherited from the division of the underlying Hilbert space or configuration space. It is shown that this partition leads to the definition of an inhomogeneous continuity equation for generic, non-local observables. This formalism is employed to describe the local evolution of the von Neumann entropy of a system of independent quantum particles out of equilibrium. Crucially, we find that all local fluctuations in the entropy are governed by an entropy current operator, implying that the production of entanglement entropy is not measured by this partitioned entropy. For systems linearly perturbed from equilibrium, it is shown that this entropy current is equivalent to a heat current, provided that the system-reservoir coupling is partitioned symmetrically. Finally, we show that any other partition of the coupling leads directly to a divergence of the von Neumann entropy. Thus, we conclude that Hilbert-space partitioning is the only partition of the von Neumann entropy that is consistent with the laws of thermodynamics.

Entanglement in flavored scalar scattering

We investigate quantum entanglement in high-energy 2 → 2 scalar scattering, where the scalars are characterized by an internal flavor quantum number acting like a qubit. Working at the 1-loop order in perturbation theory, we build the final-state density matrix as a function of the scattering amplitudes connecting the initial to the outgoing state. In this construction, the unitarity of the S-matrix is guaranteed at the required order by the optical theorem. We consider the post-scattering entanglement between the momentum and flavor degrees of freedom of the final-state particles, as well as the entanglement of the two-qubit flavor subsystem. In each case we identify the couplings of the scalar potential that can generate, destroy, or transfer entanglement between different bipartite subspaces of the Hilbert space.

Floquet systems with continuous dynamical symmetries: characterization, time-dependent noether charge, and solvability

We study quantum Floquet (periodically-driven) systems having continuous dynamical symmetry (CDS) consisting of a time translation and a unitary transformation on the Hilbert space. Unlike the discrete ones, the CDS strongly constrains the possible Hamiltonians H(t) and allows us to obtain all the Floquet states by solving a finite-dimensional eigenvalue problem. Besides, Noether’s theorem leads to a time-dependent conservation charge, whose expectation value is time-independent throughout evolution. We exemplify these consequences of CDS in the seminal Rabi model, an effective model of a nitrogen-vacancy center in diamonds without strain terms, and Heisenberg spin models in rotating fields. Our results provide a systematic way of solving for Floquet states and explain how they avoid hybridization in quasienergy diagrams.

Sample path moderate deviations for shot noise processes in the high intensity regime

We study the sample-path moderate deviation principle (MDP) for shot noise processes in the high intensity regime. The shot noise processes have a renewal arrival process, non-stationary noises (with arrival-time dependent distributions) and a general shot response function of the noises. The rate function in the MDP exhibits a memory phenomenon in this asymptotic regime, which is in contrast with that in the conventional time–space scaling regime. To prove the sample-path MDP, we first establish that this is equivalent to establishing the sample-path MDP of another process that is easier to study. We prove its finite-dimensional MDP and then establish the exponential tightness under the Skorohod J1 topology. This results in the sample-path MDP in D under the Skorohod J1 topology with a rate function that is derived from the rate function in the finite-dimensional MDP using the tools of reproducing kernel Hilbert space. In the proofs, because of the non-stationarity of shot noise process, we establish a new exponential maximal inequality and use it to prove exponential tightness and the aforementioned equivalence.

Digitizing lattice gauge theories in the magnetic basis: reducing the breaking of the fundamental commutation relations

We present a digitization scheme for the lattice SU(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{SU}}(2)$$\\end{document} gauge theory Hamiltonian in the magnetic basis, where the gauge links are unitary and diagonal. The digitization is obtained from a particular partitioning of the SU(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{SU}}(2)$$\\end{document} group manifold, with the canonical momenta constructed by an approximation of the Lie derivatives on this partitioning. This construction, analogous to a discrete Fourier transform, preserves the spectrum of the electric part of the Hamiltonian and the canonical commutation relations exactly on a subspace of the truncated Hilbert space, while the residual subspace can be projected above the cutoff of the theory.

Gauged permutation invariant matrix quantum mechanics: partition functions

The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of SN symmetric group elements U acting as X → UXUT. Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with U(N) symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of N of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.

Markovian dynamics for a quantum/classical system and quantum trajectories

Quantum trajectory techniques have been used in the theory of open systems as a starting point for numerical computations and to describe the monitoring of a quantum system in continuous time. We extend this technique to develop a general approach to the dynamics of quantum/classical hybrid systems. By using two coupled stochastic differential equations, we can describe a classical component and a quantum one which have their own intrinsic dynamics and which interact with each other. A mathematically rigorous construction is given, under the restriction of having a Markovian joint dynamics and of involving only bounded operators on the Hilbert space of the quantum component. An important feature is that, if the interaction allows for a flow of information from the quantum component to the classical one, necessarily the dynamics is dissipative. We show also how this theory is connected to a suitable hybrid dynamical semigroup, which reduces to a quantum dynamical semigroup in the purely quantum case and includes Liouville and Kolmogorov–Fokker–Planck equations in the purely classical case. Moreover, this semigroup allows to compare the proposed stochastic dynamics with various other proposals based on hybrid master equations. Some simple examples are constructed in order to show the variety of physical behaviors which can be described; in particular, a model presenting hidden entanglement is introduced.

Time-Dependent Multireference Coupled-Cluster Method (TDMRCCM) for the Bath-Dynamics in System-Bath Approach to Nonadiabatic Dynamics.

The time-dependent multireference coupled-cluster method (TDMRCCM) fits well in the scheme of the system-bath separation used to study the nonadiabatic dynamics. In TDMRCCM, a projection operator is defined as one that projects the full Hilbert space onto the space spanned by the collection of system degrees of freedom, called the model space. The inverse of this projection operator is a wave operator that acts on the model space and takes its projection back to the full wave function. This wave operator is defined as an exponential of the excitation terms and, hence, can be expanded into a Taylor series, which we have truncated in this work at the second-order of excitations. The attraction of using TDMRCCM for describing the bath dynamics is due to the exponential nature of the ansatz used in the method, which makes it possible for the higher-order excitations to be absorbed by the lower-order terms, even upon truncating the series. This improves the accuracy of the numerical calculations using TDMRCCM as an approximation for the bath-mode dynamics in the system-bath framework for nonadiabatic dynamics. We present the theoretical details of TDMRCCM and the numerical results for implementing this method to study the dynamics in the butatriene cation.

Quantum nature of spacetime near the black hole singularity

The concept of spacetime loses its usual interpretation at the essential singularity of a black hole. In consequence, all laws of physics must fail at this classical singularity. This unphysical behavior of spacetime at the singularity originates from general relativity. In order to have a consistent description of spacetime, this singularity must disappear in a quantum mechanical description of spacetime which is expected to be given by a quantum theory of gravity. In this paper, we therefore attempt to describe the quantum nature of spacetime in the vicinity of the (classical) singularity of a black hole. We take the Kantowsi–Sachs representation for the interior spacetime of a black hole and include inevitable vacuum fluctuations of matter field in the Klein–Gordon representation. Hence we obtain the Wheeler–DeWitt equation for the black hole interior and solve this equation exactly yielding a general expression for the interior wave function of the black hole. Admissible wave functions consistent with the DeWitt boundary condition implies that the Hilbert space has three nonoverlapping sectors distinguished by the relative character of the eigenvalues. Regular quantum black holes with admissible and well-behaved wave function having no singularity can exist only in two of those sectors. However, the remaining sector does not contain any regular quantum black hole.

Complexity of Digital Quantum Simulation in the Low-Energy Subspace: Applications and a Lower Bound

Digital quantum simulation has broad applications in approximating unitary evolution of Hamiltonians. In practice, many simulation tasks for quantum systems focus on quantum states in the low-energy subspace instead of the entire Hilbert space. In this paper, we systematically investigate the complexity of digital quantum simulation based on product formulas in the low-energy subspace. We show that the simulation error depends on the effective low-energy norm of the Hamiltonian for a variety of digital quantum simulation algorithms and quantum systems, allowing improvements over the previous complexities for full unitary simulations even for imperfect state preparations due to thermalization. In particular, for simulating spin models in the low-energy subspace, we prove that randomized product formulas such as qDRIFT and random permutation require smaller Trotter numbers. Such improvement also persists in symmetry-protected digital quantum simulations. We prove a similar improvement in simulating the dynamics of power-law quantum interactions. We also provide a query lower bound for general digital quantum simulations in the low-energy subspace.

Beyond a Richardson-Gaudin Mean-Field: Slater-Condon Rules and Perturbation Theory.

Richardson-Gaudin states provide a basis of the Hilbert space for strongly correlated electrons. In this study, optimal expressions for the transition density matrix elements between Richardson-Gaudin states are obtained with a cost comparable with the corresponding reduced density matrix elements. Analogues of the Slater-Condon rules are identified based on the number of near-zero singular values of the RG state overlap matrix. Finally, a perturbative approach is shown to be close in quality to a configuration interaction of Richardson-Gaudin states while being feasible to compute.

Optimal quadrature formulas for approximating strongly oscillating integrals in the Hilbert space [formula omitted] of periodic functions

The present paper is dedicated to a variational method for the construction of optimal quadrature formulas in the sense of Sard in the Hilbert space W˜2(m,m−1) of complex-valued and periodic functions. In this, the coefficients of the optimal quadrature formula are found separately in the case ωh is integer and non-integer cases. In addition, using the constructed optimal quadrature formula, the numerical results of exponentially weighted integrals of certain functions in the case m=2 is presented. The numerical results show that the order of convergence of the optimal quadrature formula is O1N+|ω|2 in the space W˜2(2,1).

A Golden Ratio Algorithm With Backward Inertial Step For Variational Inequalities

In this paper we study the convergence analysis of a Golden Ratio Algorithm with a backward inertial step and a fully adaptive step size procedure for the purpose of approximating solutions of variational inequalities in Hilbert spaces. We present a weak convergence result when the operator is quasi-monotone and locally Lipschitz continuous, and a strong convergence result in the setting of strong pseudo-monotonicity. Our algorithm features one operator evaluation and one projection computation at the current iteration. We recover interesting algorithms in the literature, for instance, the Golden Ratio Algorithm. Numerical experiments were conducted to validate the theoretical convergence analysis.

A deterministic qudit-dependent high-dimensional photonic quantum gate with weak cross-kerr nonlinearity

High-dimensional quantum systems expand the Hilbert space, enabling the processing of more intricate information and the execution of wider quantum operations. The development of high-dimensional quantum systems relies significantly on the implementation of high-dimensional gates. In the paper, we propose a deterministic 4 × 4-dimensional controlled-not (CNOT) gate for a three-photon system, where encoding on the polarization-spatial state of a single photon acts as a 4-dimensional control qudit, meanwhile the polarized state of the remaining two photons serves as a 4-dimensional target qudit. The implementation of the gate leverages weak cross-kerr nonlinearities and X-homodyne detectors, which impose lower requirements on the interaction strength and demonstrate robustness against photon loss. In accordance with the measurement of the coherent state, the relevant classical feed-forward operations are employed to make high-dimensional CNOT gate effective. Additionally, the proposed scheme demonstrates a high successful probability as a result of the utilization of mature measurement methods and straightforward optical elements. Further, the fidelity of the CNOT gate is robust against photon loss and is approximate to 1 with the existing technology. Consequently, our deterministic protocol presents a promising approach for the practical realization of high-dimensional quantum computation for photon systems.

Stability of Fixed Points of Partial Contractivities and Fractal Surfaces

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces.

A Method with Double Inertial Type and Golden Rule Line Search for Solving Variational Inequalities

In this work, we study a new line-search rule for solving the pseudomonotone variational inequality problem with non-Lipschitz mapping in real Hilbert spaces as well as provide a strong convergence analysis of the sequence generated by our suggested algorithm with double inertial extrapolation steps. In order to speed up the convergence of projection and contraction methods with inertial steps for solving variational inequalities, we propose a new approach that combines double inertial extrapolation steps, the modified Mann-type projection and contraction method, and the line-search rule, which is based on the golden ratio (5+1)/2. We demonstrate the efficiency, robustness, and stability of the suggested algorithm with numerical examples.

Loss-Assisted Anomalous Hong-Ou-Mandel Interference Based on Nonunitary Multilayer Graphene.

Hong-Ou-Mandel interference is an intrinsic quantum phenomenon that goes beyond the possibilities of classical physics, and enables numerous applications in quantum information science. While the photon-photon interaction is fundamentally limited to the bosonic nature of photons and the restricted phase responses from commonly used unitary optical elements, we present that a nonunitary material provides an alternative degree of freedom to control the two-photon quantum interference, even revealing anomalous quantum interference paths that do not exist in a unitary configuration. An elaborate lossy multilayer graphene that can work as a nonunitary beam splitter is used to explore its tunability over the effective photon-photon interaction in spatial modes, and to verify the particle exchange statistics by its experimental implementation in quantum state filter. This scheme is further extended to observe four-dimensional quantum interference patterns on the lossless and lossy beam splitters, and thus show its applicability even in higher-dimensional Hilbert space.